Art, it's NOT always the thermopile. Some visual inspecting and use of some test equipment is in order here.

What's the general condition of the hi-limit's bulb & capillary routed up there inside the fryvat? If it looks banged about and abused, replace it. It's been compromised and may have leaked a little of the gas in it that makes it work. I must point out that a hi-limit is designed to FAIL SAFE. In other words, if its functional element (the gas charge) escapes, then it fails OPEN. A v-e-r-y small leak is a hint of that feature, by it tripping at lower temperatures than it's designed to.

Is the hi-limit bulb mounted properly? That's VERY important. If it's not and has nestle its way directly up against a burner tube, then it's merely doing its job - SENSING an overheat condition.

Otherwise...

With regards the 200-250ish degree temps you stated, what tests have you performed? Is the temperature you said it heats to -

the SET POINT of the operating thermostat?

or the ACTUAL temperature?

You need be monitoring the ACTUAL temperature with a quality pyrometer/digital thermometer and, when the pilot goes out prematurely, check that hi-limit with a voltmeter.

If the hi-limit tests as open at a temp wellbelow its designed trip point (450°F), then the hi-limit needs to be replaced. If that's the case - and the sensing bulb/capillary doesn't appear compromised, then I suggest replacing the operating thermostat too. The operating thermostat is a mechanical device and wears out over time. It might be sticking occasionally and causing an overheat condition - thereby giving the hi-limit (also a mechanical device) a work out to the point that its gotten weak and has failed.

I also advise that you check your electrical connections. ANY poor connection in a 750mV circuit (what the thermopile generates) is a weak point in the circuit. 750mV voltage isn't enough potential to punch its way through a bad connection.

If you've done these things (connections are good, bulb/capillary is good and hi-limit not testing as open, other strategic voltage test points show as good), THEN you can suspect the gas valve as the culprit.

Well, criminy! Long day yesterday fixing broken stuff and forgot to mention some other basic factors that could be affecting your fryer. My being a technician, I tend to troubleshoot visually and instinctively, so I'm not used to explaining things with such detail.

Your problem may not be related to temperature at all. If there's poor gas flow/pressure to the fryer or pilot flame is weak due to blockage, the thermopile output can drop to a point where it can no longer sustain the pilot valve. It might be satisfactory when only the pilot is lit, but demands of the main burner may substantially drop gas flow to the pilot if one of these issues exists.

So, some other things I'd check:

Gas Supply - You should have no less than 7"WC (natural) or 11"WC (LP) to the fryer during its full operation. If not, then there's a gas volume problem somewhere that's restricting gas flow (ball valve partially closed, poorly coupled quick-disconnect hose connection), or a faulty line regulator. Observing the pilot flame may give a hint to these possibilities. If pilot flame intensity drops noticeably -and remains that way- when the main burner comes on, then there's an issue with gas flow or pressure to the fryer.

Pilot flame quality - Otherwise, the pilot burner might be dirty. For it to work as designed, it mixes air with gas at about a 10:1 ratio. Over time, particulates in the air can build up around the pilot burner's orifice. Giving the pilot burner a good, solid blast with compress air will usually clear that out.